Mathematical Sciences: Numerical Methods for Large-Scale Ocean Modeling

希格登研究了描述大规模海洋环流的偏微分方程组的数值解。主要的研究主题是控制方程的算子分裂。这些方程允许波速变化很大，这种情况给数值算法的稳定性和效率带来了严重的问题。解决这些问题的一种常见方法是将快动态和慢动态分成不同的子问题。快速运动基本上与深度无关，尝试用二维方程系统来模拟这些运动是很自然的。然后，将用三维系统对慢动作进行建模。然而，如果分裂是不精确的，那么假设为慢运动建模的方程实际上可能在某种程度上允许快速运动。于是出现了这样一种可能性，即这些快速运动可能会导致计算算法中的数值不稳定。这个项目的主要目标是开发一种稳定、高效、准确的控制方程分裂算子。次要目标是开发有限区域模型的开放边界条件，并研究用于空间离散的高阶差分法。当今的重大科学挑战之一是获得多年来全球海洋环流的高分辨率计算机模拟。这样的模拟非常有趣，因为海洋是地球气候系统的重要组成部分，因此对可能的全球变化的研究必须包括对海洋作用的仔细研究。例如，海洋有很大的输送热能的能力；这种输送一部分是由于主要洋流系统的大宗输送，另一部分是由于洋流产生的较小尺度的涡流造成的混合。关于这些过程的全面数据很难收集，似乎要准确理解地球气候系统，将需要基于海洋一般环流的涡旋解析模型的数值模拟。这样的模拟将带来巨大的计算需求。该项目的目标是为开发适用于这些计算的数值方法做出贡献。这项工作是与俄勒冈州立大学大气和海洋科学学院物理海洋学建模小组的成员合作进行的。

Higdon The investigator studies the numerical solution of partial differential equations that describe large-scale ocean circulation. The main topic of investigation is operator splitting for the governing equations. These equations admit wave velocities having widely varying magnitudes, and this situation creates serious problems with the stability and efficiency of numerical algorithms. One common approach to resolving these problems is to split the fast and slow dynamics into separate subproblems. The fast motions are essentially independent of depth, and it is natural to try to model these motions with a two-dimensional system of equations. The slow motions would then be modeled with a three-dimensional system. However, if the splitting is inexact, then the equations that supposedly model the slow motions may actually admit fast motions, to some degree. There then arises the possibility that these fast motions might cause numerical instabilities in the computational algorithm. The main goal of this project is to develop a stable, efficient, and accurate operator splitting for the governing equations. Secondary goals are to develop open boundary conditions for limited-area models and to examine higher-order difference methods for spatial discretizations. One of the major scientific challenges of the present day is to obtain high-resolution computer simulations of global ocean circulation over periods of many years. Such simulations are of great interest because the ocean is a crucial component of the earth's climate system, so studies of possible global change must include a careful examination of the role of the ocean. For example, the ocean has a great capacity to transport heat energy; part of this transport is due to bulk transport by major current systems, and part is due to mixing caused by smaller-scale eddies that are shed from currents. Comprehensive data on these processes are difficult to collect, and it appears that an accurate understanding of the earth's climate system will require numerical simulations based on an eddy-resolving model of the general circulation of the ocean. Such simulations will present great computational demands. The goal of this project is to contribute to the development of suitable numerical methods for these computations. This work is being performed in collaboration with members of the physical oceanography modeling group in the College of Atmospheric and Oceanic Sciences at Oregon State University.